A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below.

A simple example is the 2-dimensional Euclidean spaceR2 equipped with the Euclidean norm. Elements in this vector space (e.g., (3, 7)) are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at the origin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude.

A vector space on which a norm is defined is called a normed vector space. Similarly, a vector space with a seminorm is called a seminormed vector space. It is often possible to supply a norm for a given vector space in more than one way.

By the first axiom, absolute homogeneity, we have p(0) = 0 and p(−v) = p(v), so that by the triangle inequality

p(v) ≥ 0 (non-negativity).

A seminorm on V is a function p : V → R with the properties 1. and 2. above.

Every vector space V with seminorm p induces a normed space V/W, called the quotient space, where W is the subspace of V consisting of all vectors v in V with p(v) = 0. The induced norm on V/W is clearly well-defined and is given by:

p(W + v) = p(v).

Two norms (or seminorms) p and q on a vector space V are equivalent if there exist two real constants c and C, with c > 0 such that

If a norm p : V → R is given on a vector space V then the norm of a vector v ∈ V is usually denoted by enclosing it within double vertical lines: ‖v‖ = p(v). Such notation is also sometimes used if p is only a seminorm.

For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation | v | with single vertical lines is also widespread.

In Unicode, the codepoint of the "double vertical line" character ‖ is U+2016. The double vertical line should not be confused with the "parallel to" symbol, Unicode U+2225 ( ∥ ). This is usually not a problem because the former is used in parenthesis-like fashion, whereas the latter is used as an infix operator. The double vertical line used here should also not be confused with the symbol used to denote lateral clicks, Unicode U+01C1 ( ǁ ). The single vertical line | is called "vertical line" in Unicode and its codepoint is U+007C.

This gives the ordinary distance from the origin to the point x, a consequence of the Pythagorean theorem. The Euclidean norm is by far the most commonly used norm on Rn, but there are other norms on this vector space as will be shown below. However all these norms are equivalent in the sense that they all define the same topology.

This formula is valid for any inner product space, including Euclidean and complex spaces. For Euclidean spaces, the inner product is equivalent to the dot product. Hence, in this specific case the formula can be also written with the following notation:

The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean planeR2. This identification of the complex number x + iy as a vector in the Euclidean plane, makes the quantity x2+y2{\displaystyle {\sqrt {x^{2}+y^{2}}}} (as first suggested by Euler) the Euclidean norm associated with the complex number.

The name relates to the distance a taxi has to drive in a rectangular street grid to get from the origin to the point x.

The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope of dimension equivalent to that of the norm minus 1. The Taxicab norm is also called the ℓ{\displaystyle \ell }1 norm. The distance derived from this norm is called the Manhattan distance or ℓ{\displaystyle \ell }1 distance.

This definition is still of some interest for 0 < p < 1, but the resulting function does not define a norm,[2] because it violates the triangle inequality. What is true for this case of 0 < p < 1, even in the measurable analog, is that the corresponding Lp class is a vector space, and it is also true that the function

(without pth root) defines a distance that makes Lp(X) into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory, and harmonic analysis. However, outside trivial cases, this topological vector space is not locally convex and has no continuous nonzero linear forms. Thus the topological dual space contains only the zero functional.

In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurable functions and for the F-space of sequences with F–norm (xn)↦∑n2−nxn/(1+xn){\displaystyle (x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}}, which is discussed by Stefan Rolewicz in Metric Linear Spaces.[3] Here we mean by F-norm some real-valued function ∥⋅∥{\displaystyle \lVert \ \cdot \ \rVert } on an F-space with distance d, such that ∥x∥=d(x,0){\displaystyle \lVert x\rVert =d(x,0)}. One should note that the F-norm described above is not a norm in the usual sense because it lacks the required homogeneity property.

In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance, which is important in coding and information theory. In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero. However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous.

In signal processing and statistics, David Donoho referred to the zero"norm" with quotation marks. Following Donoho's notation, the zero "norm" of x is simply the number of non-zero coordinates of x, or the Hamming distance of the vector from zero. When this "norm" is localized to a bounded set, it is the limit of p-norms as p approaches 0. Of course, the zero "norm" is not truly a norm, because it is not positive homogeneous. Indeed, it is not even an F-norm in the sense described above, since it is discontinuous, jointly and severally, with respect to the scalar argument in scalar–vector multiplication and with respect to its vector argument. Abusing terminology, some engineers[who?] omit Donoho's quotation marks and inappropriately call the number-of-nonzeros function the L0 norm, echoing the notation for the Lebesgue space of measurable functions.

In 2D, with A a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. In 2D, each A applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a parallelogram of a particular shape, size and orientation. In 3D this is similar but different for the 1-norm (octahedrons) and the maximum norm (prisms with parallelogram base).

The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm the unit circle in R2 is a square, for the 2-norm (Euclidean norm) it is the well-known unit circle, while for the infinity norm it is a different square. For any p-norm it is a superellipse (with congruent axes). See the accompanying illustration. Due to the definition of the norm, the unit circle must be convex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and p≥1{\displaystyle p\geq 1} for a p-norm).

In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence of vectors {vn}{\displaystyle \{v_{n}\}} is said to converge in norm to v{\displaystyle v} if ∥vn−v∥→0{\displaystyle \left\|v_{n}-v\right\|\rightarrow 0} as n→∞{\displaystyle n\to \infty }. Equivalently, the topology consists of all sets that can be represented as a union of open balls.

Two norms ‖•‖α and ‖•‖β on a vector space V are called equivalent if there exist positive real numbers C and D such that for all x in V

If the vector space is a finite-dimensional real or complex one, all norms are equivalent. On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent.

Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic.

If X{\displaystyle X} and Y{\displaystyle Y} are normed spaces and u:X→Y{\displaystyle u:X\to Y} is a continuous linear map, then the norm of u{\displaystyle u} and the norm of the transpose of u{\displaystyle u} are equal.[4]

Suppose now that (p) contains a single p: since (p) is separating, p is a norm, and A = {p < 1} is its open unit ball. Then A is an absolutely convex bounded neighbourhood of 0, and p = pA is continuous.

The converse is due to Kolmogorov: any locally convex and locally bounded topological vector space is normable. Precisely:

If V is an absolutely convex bounded neighbourhood of 0, the gauge gV (so that V = {gV < 1}) is a norm.

There are several generalizations of norms and semi-norms. If p is absolute homogeneity but in place of subadditivity we require that

2′.

there is a b≥1{\displaystyle b\geq 1} such that p(u+v)≤b(p(u)+p(v)){\displaystyle p(u+v)\leq b(p(u)+p(v))} for all u,v∈V{\displaystyle u,v\in V}

then p satisfies the triangle inequality but is called a quasi-seminorm and the smallest value of b for which this holds is called the multiplier of p; if in addition p separates points then it is called a quasi-norm.

On the other hand, if p satisfies the triangle inequality but in place of absolute homogeneity we require that

1′.

there exists a k such that 0<k≤1{\displaystyle 0<k\leq 1} and for all v∈V{\displaystyle v\in V} and scalars λ{\displaystyle \lambda }: p(λv)=|λ|kp(v){\displaystyle p(\lambda v)=\left|\lambda \right|^{k}p(v)}

then p is called a k-seminorm.

We have the following relationship between quasi-seminorms and k-seminorms:

Suppose that q is a quasi-seminorm on a vector space X with multiplier b. If 0<k<log22⁡b{\displaystyle 0<k<\log _{2}^{2}b} then there exists k-seminorm p on X equivalent to q.